The Mathematics of Financial Modelingand Investment Management

20-Term Structure Page 619 Wednesday, February 4, 2004 1:33 PM

Term Structure Modeling and Valuation of Bonds and Bond Options 619

P = C C CM+

----------------------+ ----------------------+ ...+ -------------------------

( 1 + z 1 )

1

( 1 + z 2 )

2

( 1 + zN)

N

where zi is the spot rate relative to the i-th period. The coefficients

D^1

i =^ --------------------

( 1 + zi)

i

are called the discount function or discount factors.

In continuous time, as it will be demonstrated in the below, if short-

term interest rates are constant, the bond valuation formula is

C C CM+

P = -----------+ -----------+ ...+ ----------------

1 ×i 2 ×i Ni×

e e e

If short-term rates are variable, the formula is:

• ∫^1 is()sd –∫^2 is() sd –∫Nis()sd
  P = Ce +
  0

• Ce
  0


• ...+ (CM)e
  0

To consider bond valuation in continuous time, we will use many

relationships related to yield and interest rates in a stochastic environ-

ment. We begin by explicitly computing a number of these relationships

in a deterministic environment (that is, assuming that interest rates are a

known function of time) then extending these relationships to a stochas-

tic environment.

In the case of a zero-coupon bond, the financial principles of valua-

tion are those illustrated earlier when we considered very small time

intervals, in the limit infinitesimal time interval. We denote by T the

time of maturity of a bond. At a point in time s < T the time to maturity

is t = T – s. In the infinitesimal interval dt, the bond value P(t) changes

by an amount dP according to the following equation:

dP = –iPdt

where i is the deterministic short-term interest rate.

If M is the principal to be repaid at maturity, we have the initial condi-

tion M = P(0). The solution of this an ordinary differential equation with

separable variables whose solution is